Simplified model of intrinsically bursting neurons

Reviewer #1 (Public review):

Summary:

The authors present a simplified neural bursting model with explicitly controllable parameterization of oscillator dynamics designed for neural circuit modeling involved in rhythm generation.

Strengths:

(1) The purpose of the model and applied abstractions are well articulated and justified (2D model, independent parameter control).

(2) Explicit control of burst duration, inter-burst interval, amplitude, resetting-behavior/entrainment. This allows modelers to focus on circuit interactions and is especially useful when details of intrinsic currents and bursting mechanisms are unknown. One could even imagine a scenario where this model would help identify predictions on key underlying burst generation mechanisms.

(3) The model is well described and validated with simulations and comparisons to the base model and one alternative model.

(4) Circuit-level validation is convincing, as it reproduces not only trivial examples.

(5) The underlying mechanism in phase space is well reasoned and justified, extends previous work, e.g., by McKean, by improving usability.

Weaknesses:

(1) The paper heavily relies on numerical demonstrations but does not provide a formal analysis of stability, bifurcations, or entrainment. While appropriate for the intended purposes, a more formal footing could strengthen the model.

(2) Lots of nice demonstrations are shown, but it is less clear how model parameterization was chosen, how behavior depends on parameterization, and in what parameter ranges certain behavior can be expected. A more detailed description of parameterization/exploration of parameter space would greatly benefit anyone using this model in the future.

(3) Some claims on reproduction of prior locomotor CPG model and production of "more biologically realistic activity" by the presented model are overstated. The key feature of the locomotor CPG models cited was that they not only reproduced speed-dependent gait expression of intact mice, but also changes of gait expression after silencing/removal of specific commissural and long propriospinal interneurons (e.g., selective loss of trot after deleting of V0V; changes in gait expression and step-to-step variability after silencing of descending long-propriospinal neurons or ascending V3 LPNs). While likely (at least partially) feasible with the model formulation, the correspondence of these silencing/ablation of neuron classes has not been shown by the model. Importantly, though, it appears that authors didn't show how the model in general behaves under the influence of noise, which is key to reproducing LPN silencing.

Reviewer #2 (Public review):

Summary:

The authors propose a reduced model for intrinsically bursting neurons. The model simply consists of exponential decay of an adaptation variable in a phenomenological silent phase, an exponential growth of that variable in an active phase, and imposed thresholds for jumps between these phases, with some add-ons to allow for effects such as input-dependence.

Strengths:

The model could be used as a controller for an artificial system that needs to switch between on and off states with separate control of state durations. It has some flexibility to allow for variable levels of the activity variable during the active phase. The authors show that the model can be tuned to capture phase response properties of neurons and patterns generated by small networks of neurons.

Weaknesses:

The proposed approach lacks biological relevance, practicality, and originality.

(1) Biological relevance:

Central pattern generators and other bursting neurons use specific physical principles to generate their bursts of activity. These principles place constraints on the tuning of these bursts, including relationships between active and silent phase durations and other properties. By discarding these relationships, the proposed model risks losing key constraints that affect performance in biologically relevant scenarios. The proposed model does not allow for the emergence of interesting dynamical phenomena, which occur naturally in neurons and neuronal networks.

It is also important to note that spikes within bursts can be important and of interest. Biophysical models allow for easy extension to include spikes via fast sodium and potassium currents. The proposed model does not allow for such extensibility.

Finally, as shown in the seminal early-2000s work of Izhikevich, building on fast-slow decomposition work by Rinzel and others, there is a wide variety of possible neuronal bursting patterns. At the very least, several of these have been observed in neuronal recordings. The authors' model is specific to square-wave bursting.

(2) Practicality:

The model makes use of various cut-off functions and other aspects that are implemented as rules. Combining rules with differential equations makes for an awkward modeling framework that is inconvenient to implement, conceptualize, and analyze (e.g., from a bifurcation perspective). Moreover, the authors add more and more adjustments to their basic framework to capture additional features, but these add-ons simply make the model more, and unnecessarily, complicated and awkward. It's worth noting that the authors argue for their model based on the idea that more biophysical models are difficult to tune, yet they compare their model to a biophysical one that they were able to tune to achieve the various patterns that they study. They do not give any indication of how easy or hard it was to tune their own model, nor do they compare simulation times between the two models. I do note that the biophysical model seems to have 22 parameters, whereas the simplified one has 21 in Table 2, which is essentially the same number. Finally, although the authors give some extensions of the model to match observed data, their model does not seem useful for predicting performance in never-before-tested scenarios.

(3) Originality:

As the authors note, the use of low-dimensional, specifically planar, neural models dates back to early authors such as FitzHugh and Nagumo. What the authors fail to acknowledge is that Rinzel, Terman, Kopell, and others did seminal work on neuronal activity, including phenomena such as post-inhibitory rebound and fast threshold modulation, using a relaxation oscillation framework, starting several decades ago. Their work included applications to central pattern generators (e.g., see Terman and collaborators on respiratory CPGs). It is astonishing that the authors don't seem to be aware of this work and do not mention it at all. Moreover, I don't see any advantage of the proposed framework over the earlier relaxation oscillator setting, where many important mechanistic principles have already been analyzed, including extensions to networks. On a related note, even through they propose a piecewise linear model, the authors do not cite the substantial existing work on piecewise linear models (e.g., Hahnloser, Neural Networks, 1998, for an early example; 2024 SIAM Review article by Coombes et al and references therein for much more) including work specifically on bursting, nor do they cite various other previous efforts to capture bursting with simplified models including work on piecewise linear maps by Aguirre et al.

Reviewer #3 (Public review):

This computational modeling study introduces the methodology of replacing bursting neurons in a model circuit with a simplified piecewise-linear model with an "active" and a "quiet" state representing, respectively, the burst of spikes and the inter-burst interval. The shape of the active state loosely represents the intra-burst firing rate. Because (piecewise) linear systems are explicitly solvable, the transitions from quiet to active and vice versa can be calculated explicitly to match exactly what a biophysically realistic model or a biological neuron does in different conditions. The base piecewise-linear model is built to represent a 2D biophysical neuron with a cubic v-nullcline. The simplicity of the model allows for matching the kinetics of more complex models with a tractable simplified set of equations, as exemplified by approximations of burst duration and amplitude, phase-response curves, entrainment, and, finally, mimicking the activities of two CPG circuit models using this simplified representation.

Major comments

(1) The use of piecewise linear approximations to explicitly estimate properties of biophysical neurons is a well-known and common technique. This study adds nothing to the technique in terms of novelty.

(2) Although the model explicitly matches active and inactive durations of a circuit neuron, the dynamics are explicitly "clamped" by the user because the reduced model parameters explicitly depend on the input. There are cases where this is useful, for example, when we are interested in the dynamics of _other_ neurons (B, C, D, ...) within the context of activity, and we "clamp" the dynamics of neuron A. One should note that this is no better than having a look-up table. Effectively, to give a comparison, it is like using a sine wave to represent a pacemaker neuron and explicitly define its frequency at different input levels so that it responds "dynamically". However, the neuron is restricted to what the user puts in, and therefore, calling it a dynamical system is entirely wrong. I am afraid that the use of this crude tool is not described well enough in the manuscript to warn a naïve user not to fall for this trap.

(3) The phase resetting curves are used incorrectly. PRCs are useful when the perturbation is weak (soft), which would demonstrate the nature of the vector field near the limit cycle and therefore inform us of the nature of its stability or instability. A hard PRC would always reset the cycle to the fixed offset from the perturbation phase and is therefore uninformative in understanding dynamics. (It is, however, useful experimentally in identifying which neurons are part of the CPG.) The authors clearly know that the dynamics of the system away from the limit cycle do not conserve those of a biophysical neuron. So what is the point?

(4) I work on the STG, one of the systems exemplified here. Even in the small and relatively regular CPGs of the STG, the definition of the active and quiet parts of a burst is often less clear than what the authors suggest. Bursting neurons often do multiple bursts in a cycle, and therefore, substituting the burst envelope is a subjective matter. This is even more problematic in bursting neurons in the brain, where there is often no quiet period. This should be discussed.

Author response:

We thank the editors and reviewers for their time and feedback. We are encouraged by the feedback that the purpose and abstractions of the model are well articulated and justified, that the explicit control of bursting characteristics is useful, and that the circuit-level validations are convincing.

Before responding to individual reviewer comments, we would like to address the framing in the current assessment that the model "appears to have limited neurobiological relevance and utility but may be useful as a controller for an artificial system, such as in neuro-robotics applications." We respectfully suggest that this framing understates the model's relevance to neuroscience. Specifically, a growing body of literature aims to understand biological motor control by building embodied simulations. Yet, these simulations either use overly simple artificial neural network (ANN) units without dynamics or computationally intensive biophysical ones that are difficult to train. Our model is not intended as a biophysical account of how individual neurons generate bursts at the level of ionic mechanisms or spikes that goal is already well served by the conductance-based and reduced biophysical models we cite. Rather, its contribution is to make intrinsic bursting dynamics readily incorporable into neural circuit models that can be used in complex settings, with parameters that map directly onto quantities that circuit-level neuroscience most often measures and tunes in models (burst duration, duty cycle, amplitude, shape, input dependence). Indeed, Reviewer #1 notes that: "The purpose of the model and applied abstractions are well articulated and justified [...] This allows modelers to focus on circuit interactions and is especially useful when details of intrinsic currents and bursting mechanisms are unknown. One could even imagine a scenario where this model would help identify predictions on key underlying burst generation mechanisms."

We see our work as a neuroscience contribution as much as a neuro-robotics one. Bringing tractable, controllable bursting into this regime allows circuit modelers to study how intrinsic bursting interacts with circuit connectivity without committing to specific biophysical mechanisms, and it lets ANN-style models incorporate a class of dynamics that is biologically pervasive but currently underrepresented. We validated the model against two well-studied biological CPGs (the crustacean pyloric circuit and the mammalian locomotor circuit) precisely because the target use case is biological circuit modeling.

While we remain committed to the belief that bringing bio-inspired neurons with interpretable intrinsic dynamics into ANN-style modeling of biological control systems is a useful contribution as an eLife Methods paper, the reviews have made clear that we have not situated our work clearly enough within the literature. In revision, we will sharpen this positioning in the Introduction and Discussion, and better situate the model relative to both the long tradition of non-spiking relaxation-oscillator and piecewise-linear modeling in neuroscience and also to current trends in simulated control.

Public Reviews:

Reviewer #1 (Public review):

(1) Formal analysis

The paper heavily relies on numerical demonstrations but does not provide a formal analysis of stability, bifurcations, or entrainment. While appropriate for the intended purposes, a more formal footing could strengthen the model.

We agree that a formal dynamical-systems treatment would deepen the work, and we appreciate the reviewer's acknowledgment that the numerical-only approach may nevertheless be appropriate for the intended purposes. Because the model is hybrid (continuous dynamics combined with discrete switching rules), a full formal analysis is non-trivial, and we view it as a substantial follow-up rather than something to fold into the present manuscript. In revision, we will discuss more explicitly the opportunities such formal analysis presents.

(2) Parameter tuning and parameter-space characterization

It is less clear how model parameterization was chosen, how behavior depends on parameterization, and in what parameter ranges certain behavior can be expected.

We agree that this would substantially improve usability, and we will expand this aspect of the paper. The revision will include: (a) more details describing how parameters maps onto observable features of the bursting waveform, (b) recommended parameter ranges and the qualitative behaviors expected at their boundaries, and (c) practical guidance for tuning the model to match observations or embed into circuits.

(3) Locomotor CPG interneuron ablation and noise

The correspondence of these silencing/ablation of neuron classes has not been shown by the model. Importantly, though, it appears that authors didn't show how the model in general behaves under the influence of noise.

The reviewer is right that the cited work establishes validity of the circuit model in large part through silencing/ablation experiments, and we did not reproduce those experiments. We understand those gait expression phenomena to be arising from non-bursting interneuron activations and a robust solution found for connection weights between them. The half-center bursting neurons only see a time-varying input signal, and their response is well-characterized by the constant, pulse, and periodic analyses we perform. As such, we chose to reproduce a few key experiments to retain a focus on our simplified neuron model. We will rephrase the relevant passages to make this scope explicit and ensure that our reproduction claims are appropriately stated. We will also expand on how the model interfaces with noise together with the proposed parameter-space characterization.

Reviewer #2 (Public review):

(1) Biological relevance

Central pattern generators and other bursting neurons use specific physical principles to generate their bursts of activity. These principles place constraints on the tuning of these bursts, including relationships between active and silent phase durations and other properties. By discarding these relationships, the proposed model risks losing key constraints that affect performance in biologically relevant scenarios.

We agree that biophysical models impose constraints that arise from underlying mechanisms. For instance, as input alters the curved shape of nullcline-v in Figure 1, the active/quite phase durations and duty cycle change in constrained ways. The question seems to be if our model is too flexible for instance, making it too easy to achieve desired phase durations, duty cycles, and other input-dependent responses. We see this as a valuable feature of our model, not a bug. Firstly, even if our model may be expressive enough to achieve a variety of response profiles (as in Figure 3—figure supplement 3), the careful modeler will ensure matching to experimental observations. Moreover, in many circuit systems, the relevant biophysical details are often unknown for the specific neurons being modeled as noted by Reviewer #1, and the modelers' primary goal is to reproduce circuit-level activity. Such can be achieved easily with a simplified model, and also with a biophysical model as data becomes available. Finally, we should note that modelers can and do tune the parameters of biophysical models within determined ranges in order to achieve desired phase durations and duty cycles, relaxing constraints somewhat in order to reproduce appropriate activity.

It is also important to note that spikes within bursts can be important and of interest. [...] The authors' model is specific to square-wave bursting.

We agree that spikes are important and interesting in many settings, and we believe that biophysical models would be most appropriate in these cases. In many cases, too, some abstraction and simplification is desirable, and this would not necessarily detract from the model's biological relevance. As we discuss in our high-level comments, we aim to bring intrinsic bursting dynamics into the ANN-style modeling regime that typically neglects intrinsic dynamics altogether. While the simplified model may be limited in some ways, it is nevertheless useful for many common biologically relevant scenarios, as validated by our circuit experiments. Finally, we would note that many of the raised limitations (no intra-burst spike structure, restricted bursting class, abstracted constraints) are shared by the relaxation-oscillator and piecewise-linear traditions that the reviewer cites approvingly, which suggests that our model lies along a familiar abstraction continuum rather than outside it. In revision, we will explicitly acknowledge that the model captures a basic/regular form of bursting within a broader taxonomy, and clarify the conditions under which abstracting the biophysical constraints is appropriate.

(2) Practicality

The model makes use of various cut-off functions and other aspects that are implemented as rules. Combining rules with differential equations makes for an awkward modeling framework

On the modeling framework, we would defend the hybrid formulation (rules + ODE) as our aim is to prioritize usability by modelers, not the simplicity or elegance of equations. While a "pure-ODE" Fitzhugh-Nagumo-style polynomial may seem simple and elegant—with dv/dt = av^3 + bv^2 + cv + d and a, b, c, d parameters as the reviewer has pointed out a lot of complexity can arise from this. Tuning these parameters is far from intuitive, as small changes can produce nonlinear effects and qualitative shifts in behavior. Achieving the right phase durations, input-dependent scaling, waveform amplitude and shape, phase delays, and other characteristics simultaneously to match experimental data is quite cumbersome in the elegant models, not to mention the biophysical models. In contrast, these characteristics are easy to control in our model, because we translate complex dynamical behavior from implicit to explicit and surface a set of interpretable and tunable parameters.

The authors argue for their model based on the idea that more biophysical models are difficult to tune, yet they compare their model to a biophysical one that they were able to tune to achieve the various patterns that they study. They do not give any indication of how easy or hard it was to tune their own model [...] The biophysical model seems to have 22 parameters, whereas the simplified one has 21 in Table 2, which is essentially the same number.

To clarify, we did not tune the biophysical model, but rather copied its parameters from the cited work. We will make this more explicit in the relevant Methods section.

We could not simply specify or tune these parameters because they have complex biological priors that must be derived from experimental data for example, the membrane capacitance (20 pF), ionic conductance and reversal potentials (4.5 nS, -62.5 mV), and many gating kinetics parameters (slopes, midpoints, time constants for sigmoid/bell curves).

It is often the case that such parameters must be estimated in specific preparations then reused and refined over many years. For instance, the biophysical model we compare to borrowed parameters from (Kim et al. 2022), which retuned time constants relative to (Danner et al. 2017), which altered NaP conductance from (Danner et al. 2016), which retuned duty cycles from (Molkov et al. 2015), which adapted from respiratory networks of (Rubin et al. 2008), which used gating kinetics parameters from (Butera et al. 1999). Similarly, the crustacean pyloric circuit model we compare to is from (Alonso and Marder 2020), which augmented the circuit and parameters of (Prinz et al. 2004), which sampled from a database of procedurally generated parameters from (Prinz et al. 2003), which developed parameter priors from the lobster STG experimental work of (Turrigiano et al. 1995). These brief descriptions of the multi-decade lineage of parameter sets omit the substantial parallel and preceding work related their development, but they suffice to demonstrate the incredible science and effort that goes into building biophysical models for particular circuits. Such data is often unavailable and such detail is often undesirable for different research goals, in which case our simplified model is a valuable and practical tool.

The key parameters of our simplified model are observable quantities like active/quiet durations (in seconds), input-dependent duration scaling (as a fraction of intrinsic durations), input strength that induces tonic firing, etc. As such, tuning the bursting neuron parameters for circuit models was easy, with manual tuning from scratch taking less than 1 day. As Table 3 shows, the resulting parameters are often simple, elegant numbers and can be derived directly from observations. For instance, the pyloric PD active and quiet durations (200 ms and 800 ms, respectively) are set using the exact target values that (Alonso and Marder 2020) encode in their objective for a genetic algorithm to tune their model’s biophysical parameters (or rather, a subset of them for tractability).

Thus, the 22-vs-21 comparison is not very informative, because the parameters are not comparable in kind. However, to make it easier to tune our model, we will revise the manuscript to include: (a) more details describing how parameters maps onto observable features of the bursting waveform, (b) recommended parameter ranges and the qualitative behaviors expected at their boundaries, and (c) practical guidance for tuning the model to match observations or embed into circuits.

(3) Originality

What the authors fail to acknowledge is that Rinzel, Terman, Kopell, and others did seminal work on neuronal activity [...] The authors do not cite the substantial existing work on piecewise linear models [...] I don't see any advantage of the proposed framework over the earlier relaxation oscillator setting, where many important mechanistic principles have already been analyzed, including extensions to networks.

We thank the reviewer for these pointers and apologize for the gap in our literature coverage. While we had cited McKean, FitzHugh-Nagumo, Izhikevich, et al. as representative examples of different model classes, we agree that the broader relaxation-oscillator and piecewise-linear traditions deserve more comprehensive treatment including Rinzel, Terman, Kopell, et al. on relaxation-oscillators; and Hahnloser, Coombes, Aguirre, et al. on piecewise-linear models. We will expand the related work discussion and clarify how our contribution is novel and valuable.

To be clear, we do not claim to be the first to use piecewise-linear models for neurons. Our intended contribution is the specific construction a rectangular limit cycle whose horizontal/vertical decoupling permits a closed-form mapping from interpretable parameters to burst features and the demonstration that this construction integrates cleanly into firing-rate circuit models of biological CPGs, which we believe will provide realism for more complex models with learned components.

Moreover, in contrast to many other relaxation-oscillator models including the elegant Fitzhugh-Nagumo-style model we discussed above, our model is not aimed at establishing mechanistic principles or being simple enough to analyze formally. It is a practical tool that affords precise control of many bursting characteristics, which is important for closer alignment between firing-rate circuit models and biological activity. We will state this contribution more precisely in the revision so it is not conflated with a broader novelty claim.

Reviewer #3 (Public review):

(1) Novelty of piecewise-linear approximation

The use of piecewise linear approximations to explicitly estimate properties of biophysical neurons is a well-known and common technique. This study adds nothing to the technique in terms of novelty.

We agree that piecewise-linear approximations of neurons are not themselves novel, and we have not intended to claim otherwise: We cite the McKean model as a direct predecessor and, prompted by Reviewer #2, we will substantially expand citations to the relaxation-oscillator and piecewise-linear traditions (Rinzel, Terman, Kopell, Hahnloser, Coombes, Aguirre, et al.). Our intended contribution is not the use of piecewise-linear pieces per se but the specific construction: a rectangular limit cycle whose horizontal/vertical decoupling permits a closed-form, interpretable mapping from burst features (duration, duty cycle, amplitude, shape, input dependence) to dynamics, and clean integration into firing-rate circuit models of biological CPGs. We will revise the relevant passages so this contribution and the boundaries of our novelty claim are stated precisely.

(2) Dynamical system mechanism

This is no better than having a look-up table [...] The neuron is restricted to what the user puts in, and therefore, calling it a dynamical system is entirely wrong.

We would like to take the opportunity to clarify this point, because the model's behavior is much richer than the lookup-table characterization suggests. The model is closed-loop: trajectories evolve through coupled state variables whose response to time-varying input depends on current state, not on a precomputed table of input-to-output values.

Specifically:

(a) The input represents the net time-varying synaptic drive, not a clamped voltage level;

(b) The adaptation and voltage variables evolve according to coupled differential equations both on and off the limit cycle;

(c) The duration and scale parameters only constrain active/quiet durations at input endpoints (-1, 0, +1), while the response at intermediate inputs is determined by the dynamics and other parameters such as the adaptation time constant, which can qualitatively reshape the constant-input response curve (Figure 3—supplement figure 3);

(d) The response to a transient input depends on the current state for example, excitatory pulses early in the active phase have little effect, as in the biophysical model.

This is a direct result of the simplified model using a similar limit cycle and nullcline structure as the biophysical model’s dynamical system (Figure 1).

(3) PRC usage

The phase resetting curves are used incorrectly. PRCs are useful when the perturbation is weak (soft) [...] A hard PRC would always reset the cycle to the fixed offset from the perturbation phase and is therefore uninformative in understanding dynamics.

We appreciate this point and would like to clarify what we show and why. We present finite (non-infinitesimal) PRCs across a range of input strengths and signs, spanning both the "soft" (weak-perturbation) regime as well as the "hard" (strong-perturbation) regime, rather than focusing on the "hard" regime alone. Importantly, even in the strong-perturbation regime we do not see that pulses "always reset the cycle to the fixed offset from the perturbation phase". In Figure 4, we see that the active phase exhibits a non-resetting region whose size and location depend on parameters. This region governs entrainability and phase-locking offset, and is thus a key aspect of the neuron's dynamics. Moreover, the strong-perturbation regime is also biologically relevant in our circuit examples. For instance, the inhibitory connections within the pyloric CPG are strong enough to cause hard resets, and these resets shape the circuit-level dynamics we reproduce. We will revise the pulse-input section to state these points more explicitly so the rationale is clear for showing PRCs across a range of inputs.

(4) Defining active/quiet phases

The definition of the active and quiet parts of a burst is often less clear than what the authors suggest. Bursting neurons often do multiple bursts in a cycle, and therefore, substituting the burst envelope is a subjective matter. This is even more problematic in bursting neurons in the brain, where there is often no quiet period.

We agree that waveform envelope can be subjective in some preparations, and we can add this caveat to the discussion.

On neurons with no quiet period, we note that this behavior is in fact already supported in our model, as seen in Figure 3: under strong excitatory input, both the biophysical and simplified models enter a regime in which firing rate never reaches zero. As the model can generally be viewed as an abstract limit cycle that maps onto periodic waveforms through the firing function, the quiet phase need not correspond to literal silence.

On more complex waveforms, we could imagine different firing functions that produce richer burst shapes including multi-peak bursts, but we have not tried this explicitly. Of course, for research questions concerned with irregular bursting or spike-to-burst transitions, a lower-level biophysical model would be more appropriate. In revision, we will expand on how the firing function could produce more complex burst shapes.